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Lebl-Basic-Analysis-I
· Problem 7.6.2
Metric Spaces
Mathematics · fixed-point-theorem
Problem
In the proof of Picard's theorem, show that if
f
:
[
−
h
,
h
]
→
J
f: [-h,h] \to J
f
:
[
−
h
,
h
]
→
J
is continuous, then
F
(
t
,
f
(
t
)
)
F\bigl(t,f(t)\bigr)
F
(
t
,
f
(
t
)
)
is continuous on
[
−
h
,
h
]
[-h,h]
[
−
h
,
h
]
as a function of
t
t
t
. Use this to show that
T
(
f
)
(
x
)
≔
y
0
+
∫
0
x
F
(
t
,
f
(
t
)
)
d
t
T(f)(x) \coloneqq y_0 + \int_0^x F\bigl(t,f(t)\bigr)\,dt
T
(
f
)
(
x
)
:
=
y
0
+
∫
0
x
F
(
t
,
f
(
t
)
)
d
t
is well-defined and that
T
(
f
)
∈
C
(
[
−
h
,
h
]
,
R
)
T(f) \in C\bigl([-h,h],\mathbb{R}\bigr)
T
(
f
)
∈
C
(
[
−
h
,
h
]
,
R
)
.
Related theory
Wilson Fisher Fixed Point
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